Throughout the history of mathematics, one of the most
enduring challenges has been the calculation of the ratio between a
circle's circumference and diameter, which has come to be known by the
Greek letter pi.
From ancient Babylonia to the Middle Ages in
Europe to the present day of supercomputers, mathematicians have been
striving to calculate the mysterious number. They have searched for
exact fractions, formulas, and, more recently, patterns in the long
string of numbers starting with 3.14159 2653..., which is generally
shortened to 3.14. William L. Schaaf once said, "Probably no symbol
in mathematics has evoked as much mystery, romanticism, misconception
and human interest as the number pi" (Blatner, 1). We will probably
never know who first discovered that the ratio between a circle's
circumference and diameter is constant, nor will we ever know who
first tried to calculate this ratio. The people who initiated the
hunt for pi were the Babylonians and Egyptians, nearly 4000 years ago.
It is not clear how they found their approximation for pi, but one
source (Beckman) makes the claim that they simply made a big circle, and then
measured the circumference and diameter with a piece of rope. They
used this method to find that pi was slightly greater than 3, and came
up with the value 3 1/8 or 3.125 (Beckmann, 11). However, this theory
is probably a fantasy based on a
misinterpretation of the Greek word "Harpedonaptae,"
which Democritus once mentioned in a letter to a colleague. The word
literally means "rope-stretchers" or "rope-fasteners." The
misinterpretation is that these men were stretching ropes in order to
calculate circles, while they were actually making measurements in order to mark
the property limits and areas for temples, according to (Heath, 121).

A famous Egyptian piece of papyrus gives us another ancient
estimation for pi. Dated around 1650 BC, the Rhind Papyrus was written
by a scribe named Ahmes. Ahmes wrote, "Cut off 1/9 of a diameter and
construct a square upon the remainder; this has the
same area as the circle" (Blatner, 8). In other words, he implied
that pi = 4(8/9)2 = 3.16049, which is also fairly accurate. Word of
this did not spread to the East, however, as the Chinese used the
inaccurate value pi = 3 hundreds of years later.

Chronologically, the next approximation of pi is found in the Old
Testament. A fairly well known verse, 1 Kings 7:23, says: "Also he
made a molten sea of ten cubits from brim to brim, round in compass,
and five cubits the height thereof; and a line of thirty cubits did
compass it round about" (Blatner, 13). This implies that pi =
3. Debates have raged on for centuries about this verse. According
to some it was just a simple approximation, while others say that "...
the diameter perhaps was measured from outside, while the
circumference was measured from inside" (Tsaban, 76). However, most
mathematicians and scientists neglect a far more accurate
approximation for pi that lies deep within the mathematical
"code" of the Hebrew language. In Hebrew, each letter equals a
certain number, and a word's "value" is equal to the sum of its
letters. Interestingly enough, in 1 Kings 7:23, the word "line" is
written Kuf Vov Heh, but the Heh does not need to be there, and is not
pronounced. With the extra letter , the word has a value of 111, but
without it, the value is 106. (Kuf=100, Vov=6, Heh=5). The ratio of
pi to 3 is very close to the ratio of 111 to 106. In other
words, pi/3 = 111/106 approximately; solving for pi, we find pi
= 3.1415094... (Tsaban, 78). This figure is far more accurate than
any other value that had been calculated up to that point, and would
hold the record for the greatest number of correct digits for several
hundred years afterwards. Unfortunately, this little mathematical gem
is practically a secret,
as compared to the better known pi = 3 approximation.

When the Greeks took up the problem, they took two
revolutionary steps to find pi. Antiphon and Bryson of Heraclea came
up with the innovative idea of inscribing a polygon inside a circle,
finding its area, and doubling the sides over and over . "Sooner or
later (they figured), ...[there would be] so many sides that the
polygon ...[would] be a circle" (Blatner, 16). Later, Bryson also
calculated the area of polygons circumscribing the circle. This was
most likely the first time that a mathem atical result was determined
through the use of upper and lower bounds. Unfortunately, the work
boiled down to finding the areas of hundreds of tiny triangles, which
was very complicated, so their work only resulted in a few digits.
(Blatner, 16) At ap proximately the same time, Anaxagoras of
Clazomenae started working on a problem that would not be conclusively
solved for over 2000 years. After imprisonment for unlawful
preaching, Anaxagoras passed his time attempting to square the circle.
Cajori wri tes: "This is the first time, in the history of
mathematics, that we find mention of the famous problem of the
quadrature of the circle, the rock that upon which so many reputations
have been destroyed.... Anaxagoras did not offer any solution of it,
and
seems to have luckily escaped paralogisms" (Cajori 17). Since that
time, dozens of mathematicians would rack their brains trying to find
a way to draw a square with equal area to a given circle; some would
maintain that they had found methods to solve the problem, while
others would argue that it was impossible. The problem was finally
laid to rest in the nineteenth century.

The first man to really make an impact in the calculation of
pi was the Greek, Archimedes of Syracuse. Where Antiphon and Bryson
left off with their inscribed and circumscribed polygons, Archimedes
took up the challenge. However, he used a slightly dif ferent method
than they used. Archimedes focused on the polygons' perimeters as
opposed to their areas, so that he approximated the circle's
circumference instead of the area. He started with an inscribed and a
circumscribed hexagon, then doubled the si des four times to finish
with two 96-sided polygons. (Archimedes, 92) His method was as
follows...

Given a circle with radius, r = 1, circumscribe a regular
polygon A with K = 3(2n-1 sides and semiperimeter an and inscribe a
regular polygon B with K = 3(2n-1 sides and semiperimeter bn. This
results in a decreasing sequence a1, a2, a3... and an increa sing
sequence b1, b2, b3... with each sequence approaching pi. We can use
trigonometric notation (which Archimedes did not have) to find the two
semiperimeters, which are: an = K tan ((/K) and bn = K sin ((/K).
Also: an+1 = 2K tan ((/2K) and bn+1 = 2K si n ((/2K). Archimedes
began with a1 = 3 tan ((/3) = 3(3 and b1 = 3 sin ((/3) = 3(3/2 and
used 265/153 < (3 < 1351/780. He calculated up to a6 and b6 and
finally reached the conclusion that 3 10/71 < b6 < pi < a6 < 3 1/7.
Archimedes ended with a 96-sided
polygon, and numerous delicate calculations. (Archimedes, 95) The
fact that he was able to go that far and derive such a good estimation
of pi is a "stupendous feat both of imagination and calculation"
(O'Connor, 2).

For the next few hundred years, no significant breakthroughs were
made in the search for pi. Gradually, "the lead... passed from
Europe to the East" (O'Connor, 3) in the next several centuries. The
earliest value of pi used in China was 3. In 263 AD, L iu Hui
independently discovered the method used by Bryson and Antiphon, and
calculated the perimeters of regular inscribed polygons from 12 up to
192 sides, and arrived at the value pi = 3.14159, which is
absolutely correct as far as the first five digits go. Near the end
of the 5th century, Tsu Ch'ung-chih and his son Tsu Keng-chih came up
with astonishing results, when they calculated 3.1415926 < pi <
3.1415927. The father and son duo used inscribed polygons with as
many as 24,576 sides. (Blatner, 25) Soon after, the Hindu
mathematician Aryabhata gave the 'accurate' value 62,832/20,000 =
3.1416 (as opposed to Archimedes' 'inaccurate' 22/7 which was
frequently used), but he apparently never used it, nor did anyone else
for several centuries. (Beckmann, 24) Another Indian mathematician,
Brahmagupta, took a novel approach. He calculated the perimeters of
inscribed polygons with 12, 24, 48, and 96 sides as (9.65, (9.81,
(9.86, and (9.87 respectively. "And then, armed with this
information, he made the l eap of faith that as the polygons
approached the circle, the perimeter, and therefore pi, would approach
the square root of 10 [=3.162...]. He was, of course, quite wrong"
(Blatner, 26).
Although this is not as accurate as other values that
had already been calculated, it gained quite a bit of popularity as
an approximation for pi for at least a few hundred years. "Maybe
because the square root of 10 is so easy to convey and remember, this
was the value that... spread from India to Europe and was used by
mathematicians... throughout the Middle Ages" (Blatner, 26). By the
9th century, mathematics and science prospered in the Arab cultures.
It is unclear whether the Arabian mathematician, Mohammed ibn Musa
al'Khwarizmi, attempted to calculate pi, but it is clear which values
he used. He used the approximations 3 1/ 7, the square root of 10,
and 62,832/20,000. Strangely, though, the last and most accurate
value was seemingly forgotten by the Arabs and replaced by less
accurate values. (Cajori, 104)

After this, little progress
was made until a pi explosion in the end of the 16th century.
Fran&ccedille;ois Viéte, a French lawyer and amateur (but
great) mathematician, used Archimedes' method, starting with two
hexagons and doubling the number of sides sixteen times, to finish
with 393,216 sides. His final result was that 3.1415926535 <
pi < 3.1415926537. More importantly, though, Viéte became the
first man in history to describe pi using an infinite product. His
formula was: 2/pi = ((1/2)(((1/2 + 1/2 ((1/2))(((1/2 + 1/2((1/2 +
1/2(1/2))pi.... Unfortunately, this equation is not too useful
in calculating ( because it requires too many iterations before
convergence, and the square roots become quite complicated. He did
not even use his own formula in his calculation of pi.
(Beckmann, 92) Still, it was an innovative discovery that would open
many doors in the future. In 1593, Adrianus Romanus used a
circumscribed polygon with 230 sides to compute pi to 17 digits after
the decimal, of which 15 were correct. (O'Connor, 3) Just three years
later, a German named Ludolph Van Ceulen presented 20 digits, using
the Archimede an method with polygons with over 500 million sides.
Van Ceulen spent a great part of his life hunting for pi, and by the
time he died in 1610, he had accurately found 35 digits. His
accomplishments were considered so extraordinary that the digits were
cut into his tombstone in St. Peter's Churchyard in Leyden. Still
today, Germans refer to pi as the Ludolphian Number to honor the man
who had such great perseverance. (Cajori, 143) It should be noted
that up to this point, there was no symbol to signify the ratio of a
circle's circumference to its diameter. This changed in 1647 when
William Oughtred published Clavis Mathematicae and used (/( to denote
the ratio. It was not immediately embraced, until 1737, when
Leonhard Euler began using the symbol pi; then it was quickly accepted.
(Cajori, 158) In 1650, John Wallis used a very complicated method to
find another formula for pi. Basically, he approximated the area of a
quarter circle using infinitely small rectangles, and arrived at the
formula 4/pi = (3(3(5(5(7(7(9...)/(2(4(4(6(6(8(8...) which is
usually simplified to pi/2 =
(2(2(4(4(6(6(8(8...)/(1(3(3(5(5(7(7(9...). One source describes his
method as "extremely difficult and complicated" (Berggren, 292) while
another source says it is "remarkable" (Cajori, 186). Wallis showed
his formula to Lor d Brouncker, the president of the Royal Society,
who turned it into a continued fraction: pi = 4/(1 + 1/(2 + 9/(2 +
25/(2 + 49/(2 +...))))). (Cajori, 188)

In 1672, James Gregory wrote about a formula that can be used
to calculate the angle given the tangent for angles up to 45pi. The
formula is: arctan (t) = t - t3/3 + t5/5 -t7/7 + t9/9.... Ten years
later, Gottfried Leibniz pointed out that since tan ((/ 4) = 1, the
formula could be used to find pi. (Berggren, 92) Thus, one of the
most famous formulas for calculating pi was realized: (/4 = 1 - 1/3 +
1/5 - 1/7 + 1/9.... This elegant formula is one of the simplest ever
discovered to calculate pi, but it is
also fairly useless; 300 terms of the series are required to get only
2 decimal places, and 10,000 terms are required for 4 decimal places.
(O'Connor, 3) To compute 100 digits, "you would have to calculate more
terms than there are particles in the univ erse" (Blatner, 42).
However, this formula set the stage for a handful of other formulas
that would be more effective. For example, using the knowledge that
arctan (1/(3) = (/6, you can derive the following equation: arctan
(1/(3) = (/6 = 1/(3 - 1/(3(3( 3) + 1/(9(3(5) - .... After some
algebra, it simplifies to: (/6 = (1/(3)(1 - 1/(3(3) + 1/(5(32) -
1/(7(33) + 1/(9(34) -.... (O'Connor, 4) Using only six terms of this
formula, one can calculate pi = 3.141309, which isn't too far from the
real value. Sur ely, the 17th-century mathematicians were onto
something. It was just a matter of time until they discovered a
formula that was even better.

The world didn't have to wait too long, after all, before
another formula was discovered. In 1706, John Machin, a professor of
astronomy in London, armed with the knowledge that arctan x + arctan y
= arctan (x+y)/(1-xy), discovered the wonderful formula : pi/4 = 4
arctan (1/5) - arctan (1/239) = 4(1/5 - 1/(3(53) + 1/(5(55) - ...) -
(1/239 - 1/(3(2393) + 1/(5(2395) - ...). The reason that this formula
is such an improvement over the previous one is that the number 239 is
so large that we do not need very many terms of arctan (1/239) before
it converges. The other term, arctan (1/5) involves easy computations
when computing terms by hand, since it involves finding reciprocals of
powers of 5. (Blatner, 43) In fact, Machin took the initiative to
calculate p i with his new formula, and computed 100 places by hand.
(Cajori, 206) Over the next 150 years, several men used the exact same
formula to find more and more digits. In 1873, an Englishman named
William Shanks used the formula to calculate 707 places of
pi. Many years later, it was discovered that somewhere along the
line, Shanks had omitted two terms, with the result that only the
first 527 digits were correct. (Berggren, 627)

"By 1750, the number pi had been expressed by infinite
series,... its value had been computed [to over 100 digits]... and
it had been given its present symbol. All these efforts, however, had
not contributed to the solution of the ancient problem of the
quadrature of the circle" (Struik, 369). The first step was taken by
the Swiss mathematician Johann Heinrich Lambert when he proved the
irrationality of pi first in 1761 and then in more detail in
1767. (Struik, 369) His argument was, in its simplest for m, that if
x is a rational number, then tan x cannot be rational; since tan
pi/4 = 1, pi/4 cannot be rational, and therefore
pi is irrational. (Cajori, 246) Some people felt that his
proof was not rigorous enough, but in 1794, Adrien Marie Legendre gave
ano ther proof that satisfied everyone. Furthermore, Legendre also
gave the first proof that (2 is irrational. (Berggren, 297)

For the next hundred years, no major events occurred in the
pursuit of pi. More and more digits were computed, but there were no
earth-shattering breakthroughs. In 1882, Ferdinand von Lindemann
proved the transcendence of pi. (Berggren, 407) Since this means
that pi is not a solution of any algebraic equation, it lay to rest
the uncertainty about squaring the circle. Finally, after literally
thousands and thousands of lifetimes of mental toil and strain,
mathematicians finally had an absolute answer that the circle could
not be squared. Nonetheless, there are still some amateur
mathematicians today who do not understand the significance of this result, and
futilely look for techniques to square the circle.

In the twentieth century, computers took over the reigns of
calculation, and this allowed mathematicians to exceed their previous
records to get to previously incomprehensible results. In 1945, D.
F. Ferguson discovered the error in William Shanks' calc ulation from
the 528th digit onward. Two years later, Ferguson presented his
results after an entire year of calculations, which resulted in 808
digits of pi. (Berggren, 406) One and a half years later, Levi Smith
and John Wrench hit the 1000-digit-mark . (Berggren, 685) Finally, in
1949, another breakthrough emerged, but it was not mathematical in
nature; it was the speed with which the calculations could be done.
The ENIAC (Electronic Numerical Integrator and Computer) was finally
completed and funct ional, and a group of mathematicians fed in punch
cards and let the gigantic machine calculate 2037 digits in just
seventy hours. (Beckmann, 180) Whereas it took Shanks several years
to come up with his 707 digits, and Ferguson needed about one year to
g et 808 digits, the ENIAC computed over 2000 digits in less than
three days!

"With the advent of the electronic computer, there was no
stopping the pi busters" (Blatner, 51). John Wrench and Daniel Shanks
found 100,000 digits in 1961, and the one-million-mark was surpassed
in 1973. In 1976, Eugene Salamin discovered an algorith m that
doubles the number of accurate digits with each iteration, as opposed
to previous formulas that only added a handful of digits per
calculation. (Blatner, 52) Since the discovery of that algorithm, the
digits of pi have been rolling in with no end in sight. Over the past
twenty years, six men in particular, including two sets of brothers,
have led the race: Yoshiaki Tamura, Dr. Yasumasa Kanada, Jonathan and
Peter Borwein, and David and Gregory Chudnovsky. Kanada and Tamura
worked together on many
pi projects, and led the way throughout the 1980s, until the
Chudnovskys broke the one-billion-barrier in August 1989. In 1997,
Kanada and Takahashi calculated 51.5 billion (3(234) digits in just
over 29 hours, at an average rate of nearly 500,000 digits per second!
The current record, set in 1999 by Kanada and Takahashi, is
68,719,470,000 digits. (Blatner, 59) There is no knowing where or
when the search for pi will end. Certainly, the continued
calculations are unnecessary. Just thirty-nine decimal places would
be enough to compute the circumference of a circle surrounding the
known universe to within the ra dius of a hydrogen atom. (Berggren,
656) Surely, there is no conceivable need for billions of digits. At
the present time, the only tangible application for all those digits
is to test computers and computer chips for bugs. But digits aren't
really wha t mathematicians are looking for anymore. As the
Chudnovsky brothers once said: "We are looking for the appearance of
some rules that will distinguish the digits of pi from other numbers.
If you see a Russian sentence that extends for a whole page, with
hardly a comma, it is definitely Tolstoy. If someone gave you a
million digits from somewhere in pi, could you tell it was from pi? We
don't really look for patterns; we look for rules" (Blatner, 68).
Unfortunately, the Chudnovskys have also said that
no other calculated number comes closer to a random sequence of
digits. Who knows what the future will hold for the almost magical
number pi?